§6.18(i) Main Functions

For small or moderate values of x and |z|, the expansion in power series
(§6.6) or in series of spherical Bessel functions
(§6.10(ii)) can be used. For large x or |z| these series
suffer from slow convergence or cancellation (or both). However, this problem
is less severe for the series of spherical Bessel functions because of their
more rapid rate of convergence, and also (except in the case of
(6.10.6)) absence of cancellation when z=x (>0).

For large x and |z|, expansions in inverse factorial series
(§6.10(i)) or asymptotic expansions (§6.12) are
available. The attainable accuracy of the asymptotic expansions can be
increased considerably by exponential improvement. Also, other ranges of
ph⁡z can be covered by use of the continuation formulas of
§6.4.

Quadrature of the integral representations is another effective method. For
example, the Gauss-Laguerre formula (§3.5(v)) can be applied to
(6.2.2); see Todd (1954) and Tseng and Lee (1998). For
an application of the Gauss-Legendre formula (§3.5(v)) see
Tooper and Mark (1968).

Lastly, the continued fraction (6.9.1) can be used if |z| is
bounded away from the origin. Convergence becomes slow when z is near the
negative real axis, however.

§6.18(ii) Auxiliary Functions

Power series, asymptotic expansions, and quadrature can also be used to compute
the functions f⁡(z) and g⁡(z). In addition,
Acton (1974) developed a recurrence procedure, as follows. For
n=0,1,2,…, define